Optimal. Leaf size=95 \[ \frac {2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}-\frac {4 a \sqrt {c+d x}}{b^3 d^2}+\frac {x}{b^2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {371, 1398, 772} \[ \frac {2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}-\frac {4 a \sqrt {c+d x}}{b^3 d^2}+\frac {x}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sqrt {c+d x}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-c+x}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right )}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}+\frac {-a^3+a b^2 c}{b^3 (a+b x)^2}+\frac {3 a^2-b^2 c}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {x}{b^2 d}-\frac {4 a \sqrt {c+d x}}{b^3 d^2}+\frac {2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 112, normalized size = 1.18 \[ \frac {2 a^3+2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right ) \log \left (a+b \sqrt {c+d x}\right )-4 a^2 b \sqrt {c+d x}-3 a b^2 (2 c+d x)+b^3 d x \sqrt {c+d x}}{b^4 d^2 \left (a+b \sqrt {c+d x}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 163, normalized size = 1.72 \[ \frac {b^{4} d^{2} x^{2} + b^{4} c^{2} + a^{2} b^{2} c - 2 \, a^{4} + {\left (2 \, b^{4} c - a^{2} b^{2}\right )} d x - 2 \, {\left (b^{4} c^{2} - 4 \, a^{2} b^{2} c + 3 \, a^{4} + {\left (b^{4} c - 3 \, a^{2} b^{2}\right )} d x\right )} \log \left (\sqrt {d x + c} b + a\right ) - 2 \, {\left (2 \, a b^{3} d x + 3 \, a b^{3} c - 3 \, a^{3} b\right )} \sqrt {d x + c}}{b^{6} d^{3} x + {\left (b^{6} c - a^{2} b^{4}\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 102, normalized size = 1.07 \[ -\frac {\frac {2 \, {\left (b^{2} c - 3 \, a^{2}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{4} d} - \frac {{\left (d x + c\right )} b^{2} d - 4 \, \sqrt {d x + c} a b d}{b^{4} d^{2}} + \frac {2 \, {\left (a b^{2} c - a^{3}\right )}}{{\left (\sqrt {d x + c} b + a\right )} b^{4} d}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 125, normalized size = 1.32 \[ -\frac {2 a c}{\left (a +\sqrt {d x +c}\, b \right ) b^{2} d^{2}}-\frac {2 c \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d^{2}}+\frac {x}{b^{2} d}+\frac {2 a^{3}}{\left (a +\sqrt {d x +c}\, b \right ) b^{4} d^{2}}+\frac {6 a^{2} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{4} d^{2}}+\frac {c}{b^{2} d^{2}}-\frac {4 \sqrt {d x +c}\, a}{b^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 90, normalized size = 0.95 \[ -\frac {\frac {2 \, {\left (a b^{2} c - a^{3}\right )}}{\sqrt {d x + c} b^{5} + a b^{4}} - \frac {{\left (d x + c\right )} b - 4 \, \sqrt {d x + c} a}{b^{3}} + \frac {2 \, {\left (b^{2} c - 3 \, a^{2}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{4}}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 98, normalized size = 1.03 \[ \frac {x}{b^2\,d}+\frac {2\,\left (a^3-a\,b^2\,c\right )}{b\,\left (b^4\,d^2\,\sqrt {c+d\,x}+a\,b^3\,d^2\right )}-\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (2\,b^2\,c-6\,a^2\right )}{b^4\,d^2}-\frac {4\,a\,\sqrt {c+d\,x}}{b^3\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 42.38, size = 131, normalized size = 1.38 \[ \begin {cases} \frac {2 \left (- \frac {a \left (a^{2} - b^{2} c\right ) \left (\begin {cases} \frac {\sqrt {c + d x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {1}{b \left (a + b \sqrt {c + d x}\right )} & \text {otherwise} \end {cases}\right )}{b^{3} d} - \frac {2 a \sqrt {c + d x}}{b^{3} d} + \frac {c + d x}{2 b^{2} d} + \frac {\left (3 a^{2} - b^{2} c\right ) \left (\begin {cases} \frac {\sqrt {c + d x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {c + d x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{3} d}\right )}{d} & \text {for}\: d \neq 0 \\\frac {x^{2}}{2 \left (a + b \sqrt {c}\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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